LITTLE ROCK, Ark.—Vertex Pharmaceuticals Inc.'s $300,000-a-year cystic-fibrosis drug has sparked a legal battle here, where the state's Medicaid program is restricting access to the expensive therapy.

In a lawsuit filed in Arkansas federal court last month, three people suffering from the fatal lung disease allege Medicaid officials have for two years denied them access to Kalydeco because of its cost. The plaintiffs allege state...